July 7, 192 1] 



NATURE 



587 



given in this paper, and it appears that the effect 

 was observed and described by Savart as early as 

 1839. W. B. Morton. 



Queen's University, Belfast, June 21. 



An Algebraical Identity. 



The values of the coefficients of Y and Z are given 

 in Prof. Mathew's "Theory of Numbers," p, 218, 

 for the primes 3 to 31. I have calculated the values 

 for the primes 37 to 61 by the method given by Prof. 

 Mathews on p. 216 of his book. My result in the 

 case of ^ = 37 agrees with that given in Nature of 

 June 9, p. 456. The other results are as follows, the 

 coefficients being given to the middle term inclusive 

 when that exists, and in the other case to the first of 

 the pair of terms at the middle : — 



/ = 4i ; V, 2, I. II, 16, 14, 29, 30, 22, 36, 34, 20; 

 Z, I, I, 2, 4, 3, 4. 6, 4, 4, 6. 



/ = 43 ; Y, 2, I, - 10, 6, 16, -20, -4, 27, - 15, -7, 17 ; 

 Z, I, o, -2, 2, 2, -4, I, 3, -3, I. 



;) = 47; Y, 2, i, -11, -17, -9. 6, 29,37,20, 

 Z, I, I, - I, -3, -4, - 3, I, 4, 



-2, - 16, - II ; 



4. 2-, - I. 



/ = 53; Y, 2, I, 14, -6, 8, -14, -4, 19. -12, 

 Z, I, o, 2, -2, o, o, - I, 4, 



24, -9, - II, 27, - 25 ; 

 - 2, I, I, - 3, 5. 



/ = 59; Y,2,i, -14, 8,7,-35, 22,12,-33, 

 Z, I, o, -2, 3, - I, - 4, 4, - I, 



18, 23, -29, 14, 18, - 29 ; 

 - 4: 3. I. - 4, 4. - I- 



/ = 6i ; Y, 2, I, 16, - 7, 32, -20. 63, -33, 72, 



7', I, o, 3, - 2, 6, - 3, 9, - 6, 



-54, 89, -62, 88, -89, 95, -81; 



10, - 7, 12, -10, II, -II, 13. 



The first case where Legendre's rule fails is 



^ = 41. H. C. POCKLINGTON. 



5 Well Close Place, Leeds, June 12. 



The result for p = 27 given in Nature of June 9, 

 p. 456, was found to conform to Legendre's rule; 

 since this rule fails in the case of ^ = 61, it is interest- 

 ing, as noted by Prof. Mathews, to know if this is 

 the lowest prime for which the rule fails. 



I have worked out the case for p = ^i, and find the 

 expression of the 20th degree in x for Y to have the 

 following coefficients : — 



2+1 + 11 + 16 + 14-12-11-19-5-7 + 20-7-5 



-19-11-12 + 14+16+11 + 1+2. 



And since X is of the 40th degree in x, each co- 

 efficient^ being +1, I find (Y'-^X)/^i of the 38th 

 degree in x with the following coefficients : — 



I + 2 + 5 + 8+ 12 + 2- 12- 28 -20 -14 +15+ 18+ 19 

 -24-23-36 + 9 + 28 + 72 + 28 + 9-36-23-24 + 19 



+ 18 +15 -14 -20 -28 -12 + 2 + 12 + 8 + 5 + 2+ I. 



Putting x=i, the sum of these coefficients equals —4, 

 hence (Y* — 4X)/4i cannot be a square, so that the 

 rule fails, 41 being the lowest prime for failure. 



J. CULLEN, S.J. 

 Stonyhurst College, June 17. 



NO. 2697, VOL. 107] 



Artefacts and their Geological Age. 



In Nature of June 9, p. 458, Mr. J. Reid Moir 

 describes some flint implements found embedded in 

 the surface of the ferruginous "pan " at the base of 

 the cliff near Sheringham, and he bases certain con- 

 clusions upon this find. From his description of the 

 occurrence of the flints it seems clear that in this 

 case they may not be of the same age as the "pan," 

 in the upper surface of which they were found 

 embedded. A little while ago, on the beach at Flam- 

 borough, a small deposit of ferruginous conglomerate 

 was found, in which there was embedded a typical 

 Neolithic "scraper," as well as several pebbles, and 

 the conglomerate proved to be formed upon a horse- 

 shoe. But no one here assumed that the horse- 

 shoe was Neolithic in date, or that the scraper was 

 made during the past fifty years. More recently, on 

 an excursion to South Ferriby, on the Humber shore, 

 firmly embedded in a ferruginous "pan," immediately 

 at the base of a cliff of Boulder Clay, was a trouser 

 button. It had to be extracted with a hammer. But 

 no Yorkshire geologist is likely to write to Nature 

 to try to prove that pre-Glacial man in the Humber 

 district wore trousers. T. Sheppard. 



The Municipal Museum, Hull, June 27. 



iron Currency-Bars. 



In Nature of May 19, p. 372, reference is made to 

 iron currency-bars and "early British water-clocks." 

 The discovery of the true nature of the currency-bars 

 is not, as is implied, a recent one, but was made in 

 1905 by Mr. Reginald Smith (see his paper. Pro- 

 ceedings of the Society of Antiquaries, vol. xx., pp. 

 179-94). Similarly, the "early British water-clocks" 

 were first authoritatively dealt with by Mr. Smith in 

 1907 (see his paper, ibid., vol. xxi., pp. 319 sqq.). 

 I understand that Dr. Newton Friend made this 

 quite clear in the paper referred to, and gave his 

 references. 



It may be of interest to add that a hoard of cur- 

 rency-bars has recently been found near Winchester, 

 and that the site is now being excavated by a band 

 of volunteers under the direction of Mr. R. W. 

 Hooley, hon. curator of the Winchester Museum. 

 The currency-bars were exhibited at a recent meeting 

 of the Society of Antiquaries. 



June 20. O. G. S. Crawford. 



History of the Churn. 



In No. 23 of the Agricultural Ledger, issued by 

 the Government of India, and published in July, -1895, 

 there is a precis of official correspondence on the 

 Indian churn which begins : " In a letter addressed 

 to the Government of India, Herr B. Martiny, of 

 Berlin, asked for information regarding the Indian 

 Churn. He there announces that he is engaged 

 writing a history of the Churn, and is desirous of ob- 

 taining certain particulars regarding ' the old Indian 

 Churn,' of which he furnished a drawing." Has this 

 " History of the Churn " been published in book form, 

 or in the journal or transactions of any society? 

 May I ask readers of Nature if they can afford any 

 information on the subject? If Herr Martiny 

 addressed similar communications about native or 

 ancient forms of churns to European, American, and 

 Far Eastern Governments, and had his inquiry as 

 fully replied to as it was by the Government of India, 

 there must be pigeon-holed somewhere a mass of in- 

 teresting data. R. Hedger Wallace. 



June 17. 



